A note on a class of gyrogroups

In \cite{1}, a class of gyrogroups $(G,\odot )$ has been associated to certain groups $(G,\cdot )$. We give a necessary and sufficient condition for $(G,\odot )$ to be gyrocommutative. We also prove that under a suitable assumption two finite groups central by a $2$-Engel group are isomorphic if and only if their associated gyrogroups are isomorphic.